Two-point boundary value problem for systems of pseudo-differential equations under boundary conditions containing pseudo-differential operators
Abstract
This paper deals with a two-point boundary value problem for pseudodifferential equations and for systems of second order pseudodifferential equations under boundary conditions containing pseudodifferential operators. The need to consider pseudodifferential operators is caused by two reasons, first, such equations appear more and more often in applied problems, and second, by considering such equations, it is possible to achieve the well-posedness of the boundary value problem in the Schwartz space S and in its dual space.First, we consider a scalar pseudodifferential equation with a symbol belonging to the space $C_{-\infty}^{\infty}$, consists of infinitely differentiable functions of polinomial growth. For this equation it is found of the boundary condition under which a specific type the boundary value problem is well-posed in the space S. In addition, an example of a differential-difference equation and a specific boundary condition with a convolution-type pseudo-differential operator under which this boundary value problem is well-posed in the space S are given.Then we consider a system of two pseudodifferential equations with symbols from the space $C _ {-\infty} ^ {\infty}$. For this system, we prove the existence of a well posed boundary value problem in the space S. The Fourier transform and the reduction of the system to a triangular form are used in the proof. In this case, we also give an example of a system and a specific boundary condition under which this boundary value problem is correct in the space S.Thus, the work proves that for any pseudo-differential equation, as well as for a system of two pseudo-differential equations, there is always a correct boundary value problem in the $S$ space, while the boundary conditions contain pseudo-differential operators. The algorithm for constructing correct boundary conditions is also indicated. They are pseudo-differential operators whose symbols depend on the symbols of pseudo-differential equations.Downloads
References
B.J. Ptashnik., V.S. Ilkiv, I.I. Kmit, V.M. Polishchuk, Nonlocal boundary value problems for partial differential equations. 2002. Kyiv: Naukova Dumka, 415 p. (in Ukrainian). ISBN 966-00-0095-2.
A.M. Kuz, B.Yo. Ptashnyk, Problem for hyperbolic system of equations having constant coefficients with integral conditions with respect to the time variable, Carpathian Math. Publ., - 2014. 6, N. 2. - P. 282-299. DOI: 10.15330/cmp.6.2.282-299.
A. Bouziani, N. Merazga. Solution to a pseudoparabolic problem with conditions, EJDE. - 2006. - No. 115. - P. 1-18. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://math.unt.edu ftp ejde.math.txstate.edu (login: ftp).
A.A. Makarov, D.A. Levkin. Boundary-value problems in a layer for evolutionary pseudo-differential equations with integral conditions. Visnyk of V. N. Karazin Kharkiv National University, Ser. "Mathematics, Applied Mathematics and Mechanics". - 2018. - 87. - P. 61-67. DOI: 10.26565/2221-5646-2018-87-05.
A.A. Makarov. The existence of a correct two-point boundary value problem in a layer for systems of pseudo-differential equations, Differential Equations. 1994. - Vol. 30, No. 1. - P. 144-150. DOI: 10.15330/cmp.6.2.282-299.
L. R. Volevich, S. G. Gindikin. Generalized functions and equations in convolutions. M.: Fizmatlit. - 1994. - 336 p. - ISBN 5-02-014601-3.
A.A. Makarov. Parabolic boundary value problems for systems of pseudodifferential equations in an infinite layer, Differential Equations. 1996. - Vol. 32, No. 5. - P. 636-642.
Roger A. Horn, Charles R. Johnson. Matrix Analysis. Cambridge University Press. - 2013. - 643 p. -- ISBN 0521839408, 9780521839402.
Copyright (c) 2022 О.А. Макаров, І.Г. Ніколенко
![Creative Commons License](http://i.creativecommons.org/l/by-nc-nd/4.0/88x31.png)
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
The copyright holder is the author.
Authors who publish with this journal agree to the following terms:
1. Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal. (Attribution-Noncommercial-No Derivative Works licence).
2. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
3. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (see The Effect of Open Access).